Extremal problems in the function theory associated with the $n$-fold symmetry

We consider some conventional problems of the theory of functions of a complex variable such that their extremal configurations have the $n$-fold symmetry. We discuss two-point distortion theorems corresponding to the two-fold symmetry. New estimates are obtained for the module of a doubly connected domain. These estimates generalize known results by Rengel, Grötzsch, and Teichmüller to the case of rings with the $n$-fold symmetry, where $n\ge2$. New distortion theorems are proved for functions meromorphic and univalent in a disk or in a ring. In these theorems, the extremal function also has the corresponding symmetry. All of the problems mentioned above are unified by the method applied; this method is based on properties of the conformal capacity and on symmetrization.